Ear-Slicing for Matchings in Hypergraphs

We study when a given edge of a factor-critical graph is contained in a matching avoiding exactly one, pregiven vertex of the graph. We then apply the results to always partition the vertex-set of a $3$-regular, $3$-uniform hypergraph into at most one triangle (hyperedge of size $3$) and edges (subsets of size $2$ of hyperedges), corresponding to the intuition, and providing new insight to triangle and edge packings of Cornu\'ejols' and Pulleyblank's. The existence of such a packing can be considered to be a hypergraph variant of Petersen's theorem on perfect matchings, and leads to a simple proof for a sharpening of Lu's theorem on antifactors of graphs

Odd Paths, Cycles and TT-joins: Connections and Algorithms

Minimizing the weight of an edge set satisfying parity constraints is a challenging branch of combinatorial optimization as witnessed by the binary hypergraph chapter of Alexander Schrijver's book ``Combinatorial Optimization'' (Chapter 80). This area contains relevant graph theory problems including open cases of the Max Cut problem, or some multiflow problems. We clarify the interconnections of some problems and establish three levels of difficulties. On the one hand, we prove that the Shortest Odd Path problem in an undirected graph without cycles of negative total weight and several related problems are NP-hard, settling a long-standing open question asked by Lov\'asz (Open Problem 27 in Schrijver's book ``Combinatorial Optimization''. On the other hand, we provide a polynomial-time algorithm to the closely related and well-studied Minimum-weight Odd $\{s,t\}$-Join problem for non-negative weights, whose complexity, however, was not known; more generally, we solve the Minimum-weight Odd $T$-Join problem in FPT time when parameterized by $|T|$. If negative weights are also allowed, then finding a minimum-weight odd $\{s,t\}$-join is equivalent to the Minimum-weight Odd $T$-Join problem for arbitrary weights, whose complexity is only conjectured to be polynomially solvable. The analogous problems for digraphs are also considered.Comment: 24 pages, 2 figure

Boxicity and Interval-Orders: Petersen and the Complements of Line Graphs

The boxicity of a graph is the smallest dimension $d$ allowing a representation of it as the intersection graph of a set of $d$-dimensional axis-parallel boxes. We present a simple general approach to determining the boxicity of a graph based on studying its ``interval-order subgraphs''. The power of the method is first tested on the boxicity of some popular graphs that have resisted previous attempts: the boxicity of the Petersen graph is $3$, and more generally, that of the Kneser-graphs $K(n,2)$ is $n-2$ if $n\ge 5$, confirming a conjecture of Caoduro and Lichev [Discrete Mathematics, Vol. 346, 5, 2023]. Since every line graph is an induced subgraph of the complement of $K(n,2)$, the developed tools show furthermore that line graphs have only a polynomial number of edge-maximal interval-order subgraphs. This opens the way to polynomial-time algorithms for problems that are in general $\mathcal{NP}$-hard: for the existence and optimization of interval-order subgraphs of line-graphs, or of interval-completions of their complement.Comment: 17 pages, 5 figures, appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

How many matchings cover the nodes of a graph?

Given an undirected graph, are there $k$ matchings whose union covers all of its nodes, that is, a matching-$k$-cover? A first, easy polynomial solution from matroid union is possible, as already observed by Wang, Song and Yuan (Mathematical Programming, 2014). However, it was not satisfactory neither from the algorithmic viewpoint nor for proving graphic theorems, since the corresponding matroid ignores the edges of the graph. We prove here, simply and algorithmically: all nodes of a graph can be covered with $k\ge 2$ matchings if and only if for every stable set $S$ we have $|S|\le k\cdot|N(S)|$. When $k=1$, an exception occurs: this condition is not enough to guarantee the existence of a matching-$1$-cover, that is, the existence of a perfect matching, in this case Tutte's famous matching theorem (J. London Math. Soc., 1947) provides the right `good' characterization. The condition above then guarantees only that a perfect $2$-matching exists, as known from another theorem of Tutte (Proc. Amer. Math. Soc., 1953). Some results are then deduced as consequences with surprisingly simple proofs, using only the level of difficulty of bipartite matchings. We give some generalizations, as well as a solution for minimization if the edge-weights are non-negative, while the edge-cardinality maximization of matching-$2$-covers turns out to be already NP-hard. We have arrived at this problem as the line graph special case of a model arising for manufacturing integrated circuits with the technology called `Directed Self Assembly'.Comment: 10 page

Layers and Matroids for the Traveling Salesman's Paths

Gottschalk and Vygen proved that every solution of the subtour elimination linear program for traveling salesman paths is a convex combination of more and more restrictive "generalized Gao-trees". We give a short proof of this fact, as a layered convex combination of bases of a sequence of increasingly restrictive matroids. A strongly polynomial, combinatorial algorithm follows for finding this convex combination, which is a new tool offering polyhedral insight, already instrumental in recent results for the $s-t$ path TSP

Complements of nearly perfect graphs

A class of graphs closed under taking induced subgraphs is $\chi$-bounded if there exists a function $f$ such that for all graphs $G$ in the class, $\chi(G) \leq f(\omega(G))$. We consider the following question initially studied in [A. Gy{\'a}rf{\'a}s, Problems from the world surrounding perfect graphs, {\em Zastowania Matematyki Applicationes Mathematicae}, 19:413--441, 1987]. For a $\chi$-bounded class $\cal C$, is the class $\bar{C}$ $\chi$-bounded (where $\bar{\cal C}$ is the class of graphs formed by the complements of graphs from $\cal C$)? We show that if $\cal C$ is $\chi$-bounded by the constant function $f(x)=3$, then $\bar{\cal C}$ is $\chi$-bounded by $g(x)=\lfloor\frac{8}{5}x\rfloor$ and this is best possible. We show that for every constant $c>0$, if $\cal C$ is $\chi$-bounded by a function $f$ such that $f(x)=x$ for $x \geq c$, then $\bar{\cal C}$ is $\chi$-bounded. For every $j$, we construct a class of graphs $\chi$-bounded by $f(x)=x+x/\log^j(x)$ whose complement is not $\chi$-bounded

Soluble B-cell maturation antigen as a monitoring marker for multiple myeloma

Objective: Response to treatment in multiple myeloma (MM) is routinely measured by serum and urine M-protein and free light chain (FLC), as described by the International Myeloma Working Group (IMWG) consensus statement. A non-negligible subgroup of patients however present without measurable biomarkers, others become oligo or non-secretory during recurrent relapses. The aim of our research was to evaluate soluble B-cell maturation antigen (sBCMA) as a monitoring marker measured concurrent with the standard monitoring in MM patients at diagnosis, at relapse and during follow up, in order to establish its potential usefulness in oligo and non-secretory disease.Method: sBCMA levels were measured in 149 patients treated for plasma cell dyscrasia (3 monoclonal gammopathy of unknown significance, 5 smoldering myeloma, 7 plasmacytoma, 8 AL amyloidosis and 126 MM) and 16 control subjects using a commercial ELISA kit. In 43 newly diagnosed patients sBCMA levels were measured at multiple timepoints during treatment, and compared to conventional IMWG response and progression free survival (PFS).Results: sBCMA levels among control subjects were significantly lower than among newly diagnosed or relapsed MM patients [20.8 (14.7–38.7) ng/mL vs. 676 (89.5–1,650) and 264 (20.7–1,603) ng/mL, respectively]. Significant correlations were found between sBCMA and the degree of bone marrow plasma cell infiltration. Out of the 37 newly diagnosed patients who have reached partial response or better per IMWG criteria, 33 (89%) have had at least a 50% drop in sBCMA level by therapy week 4. Cohorts made similarly to IMWG response criteria—achieving a 50% or 90% drop in sBCMA levels compared to level at diagnosis—had statistically significant differences in PFS.Conclusion: Our results confirmed that sBCMA levels are prognostic at important decision points in myeloma, and the percentage of BCMA change is predictive for PFS. This highlights the great potential use of sBCMA in oligo- and non-secretory myeloma

Cadaverine, a metabolite of the microbiome, reduces breast cancer aggressiveness through trace amino acid receptors

Recent studies showed that changes to the gut microbiome alters the microbiome-derived metabolome, potentially promoting carcinogenesis in organs that are distal to the gut. In this study, we assessed the relationship between breast cancer and cadaverine biosynthesis. Cadaverine treatment of Balb/c female mice (500 nmol/kg p.o.q.d.) grafted with 4T1 breast cancer cells ameliorated the disease (lower mass and infiltration of the primary tumor, fewer metastases, and lower grade tumors). Cadaverine treatment of breast cancer cell lines corresponding to its serum reference range (100-800 nM) reverted endothelial-to-mesenchymal transition, inhibited cellular movement and invasion, moreover, rendered cells less stem cell-like through reducing mitochondrial oxidation. Trace amino acid receptors (TAARs), namely, TAAR1, TAAR8 and TAAR9 were instrumental in provoking the cadaverine-evoked effects. Early stage breast cancer patients, versus control women, had reduced abundance of the CadA and LdcC genes in fecal DNA, both responsible for bacterial cadaverine production. Moreover, we found low protein expression of E. coli LdcC in the feces of stage 1 breast cancer patients. In addition, higher expression of lysine decarboxylase resulted in a prolonged survival among early-stage breast cancer patients. Taken together, cadaverine production seems to be a regulator of early breast cancer